An easy way to catch fairies, in my opinion, is to get your bottle out when you're not too close to the fairy. Approach slowly and Z-target it as soon as you can. The best time to swing is when the tip at the bottom appears ("A Catch").

The targeting will ensure that Link gets to the right height for a successful catch. Be careful not to approach too fast, as you might miss the right timing and get instant healed instead.